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POST UTME IGBINEDION UNIVERSITY 2017 Mathematics | Objective

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Question 1

Find the volume of the solid formed by revolving the region bounded by the curves y = x^2, y = 0, and x = 2 about the x-axis.

Correct A. \frac{32\pi}{3}

B. \frac{64\pi}{3}

C. \frac{128\pi}{3}

D. \frac{256\pi}{3}

Correct Answer: A

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Question 2

If u = \sin^3 x, find \frac{du}{dx} u\sing the chain rule.

Correct A. 3\sin^2 x \cos x

B. \cos x

C. \sin x

D. \cos^3 x

Correct Answer: A

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Question 3

Find the sum of the first 10 terms of the geometric series 2 + 6 + 18 + ...

A. 104

B. 106

Correct C. 108

D. 110

Correct Answer: C

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Question 4

Solve the system of equations u\sing matrices: \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3 \\ 8 \end{bmatrix}

Correct A. x = 1, y = 2

B. x = 2, y = 1

C. x = 3, y = 4

D. x = 4, y = 3

Correct Answer: A

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Question 5

Find the area of the triangle with vertices (2, 3), (4, 5), and (6, 7).

A. 10

B. 20

Correct C. 30

D. 40

Correct Answer: C

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Question 6

Let \( S = {1, 2, 3, 4, 5} \) be a set of integers. Find the number of subsets of ( S ) that contain exactly two elements.

A. 10

Correct B. 15

C. 20

D. 25

Correct Answer: B

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Question 7

Find the sum of the first 10 terms of the arithmetic sequence ( 2, 5, 8, ldots ).

A. 50

B. 55

Correct C. 60

D. 65

Correct Answer: C

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Question 8

The polynomial ( P(x) = x^3 - 6x^2 + 11x - 6 ) has a root at \( x = 1 \). Find the other two roots.

Correct A. 2, 3

B. 3, 4

C. 4, 5

D. 5, 6

Correct Answer: A

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Question 9

The mean of the numbers 2, 4, 6, 8, and 10 is 6. Find the mean of the numbers 3, 5, 7, 9, and 11.

A. 7

Correct B. 8

C. 9

D. 10

Correct Answer: B

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Question 10

Find the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 4 \).

A. 16

B. 20

Correct C. 24

D. 28

Correct Answer: C

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Question 11

Determine the value of x in the equation \( \sin^2\( x \ \) + \cos^2(x) = 1 ) if \( \sin\( x \ \) = \frac{3}{5} ).

Correct A. \( \frac{pi}{4} \)

B. \( \frac{pi}{3} \)

C. \( \frac{pi}{2} \)

D. \( \frac{pi}{6} \)

Correct Answer: A

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Question 12

Find the derivative of the function ( f(x) = \frac{1}{x^2 + 1} ) u\sing the chain rule.

Correct A. ( f'(x) = \frac{-2x}{\( x^2 + 1 \)^2} )

B. ( f'(x) = \frac{2x}{\( x^2 + 1 \)^2} )

C. ( f'(x) = \frac{2}{\( x^2 + 1 \)^2} )

D. ( f'(x) = \frac{-2}{\( x^2 + 1 \)^2} )

Correct Answer: A

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Question 13

Solve the inequality \( 2^x > 10 \) u\sing \logarithms.

Correct A. \( x > 3.3219 \)

B. \( x < 3.3219 \)

C. \( x > 1.5 \)

D. \( x < 1.5 \)

Correct Answer: A

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Question 14

Determine the value of \( \sqrt[3]{64} \) u\sing the properties of exponents.

Correct A. ( 4 )

B. ( 8 )

C. ( 16 )

D. ( 32 )

Correct Answer: A

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Question 15

Find the probability of drawing two hearts from a s\tandard deck of 52 cards.

Correct A. \( \frac{1}{221} \)

B. \( \frac{1}{52} \)

C. \( \frac{1}{26} \)

D. \( \frac{1}{13} \)

Correct Answer: A

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Question 16

Find the area under the curve \( y = \frac{1}{2}x^2 + 3x - 2 \) from \( x = 0 \) to \( x = 4 \).

Correct A. \( \frac{1}{2} \times 4^3 + 3 \times 4^2 - 2 \times 4 \)

B. \( \frac{1}{2} \times 4^3 + 3 \times 4^2 - 2 \times 4 + \frac{1}{2} \)

C. \( \frac{1}{2} \times 4^3 + 3 \times 4^2 - 2 \times 4 - \frac{1}{2} \)

D. \( \frac{1}{2} \times 4^3 + 3 \times 4^2 - 2 \times 4 \)

Correct Answer: A

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Question 17

A histogram of exam scores has a mean of 60 and a s\tandard deviation of 10. If 75% of the scores fall below 70, what is the z-score corresponding to a score of 70?

A. 0.5

Correct B. 1

C. 1.25

D. 1.5

Correct Answer: B

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Question 18

Solve the quadratic equation \( x^2 + 4x - 5 = 0 \).

Correct A. \( x = -5, x = 1 \)

B. \( x = -1, x = 5 \)

C. \( x = -2, x = 2.5 \)

D. \( x = -1, x = 5 \)

Correct Answer: A

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Question 19

A bag contains 5 red balls and 3 blue balls. If a ball is drawn at random, what is the probability that it is blue?

A. \( \frac{1}{2} \)

Correct B. \( \frac{3}{8} \)

C. \( \frac{1}{4} \)

D. \( \frac{3}{5} \)

Correct Answer: B

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Question 20

Find the sum of the first 5 terms of the geometric sequence ( 2, 6, 18, ... ).

Correct A. ( 62 )

B. ( 64 )

C. ( 66 )

D. ( 68 )

Correct Answer: A

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Question 21

Let X and Y be indep\endent random variables with probability density functions f_X(x) = 2x, 0 < x < 1, and f_Y(y) = 3y^2, 0 < y < 1. Find the probability that X + Y < 1.

A. 1/4

B. 1/2

Correct C. 3/4

D. 1

Correct Answer: C

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Question 22

Find the determinant of the matrix \[ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]

Correct A. 0

B. 1

C. 2

D. 3

Correct Answer: A

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Question 23

Let \( \vec{a} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \( \vec{b} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \). Find the dot product of \( \vec{a} \) and \( \vec{b} \).

A. 11

Correct B. 12

C. 13

D. 14

Correct Answer: B

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Question 24

Simplify the expression \( \frac{2^3 \cdot 3^2}{2^2 \cdot 3} \).

A. 4

B. 6

Correct C. 8

D. 12

Correct Answer: C

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Question 25

Find the value of \( \log_{10} (100) \).

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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